Abstract:
This article is devoted to studying a class of generalized $p(x)$-Laplacian Kirchhoff
equations in the following form:
\begin{align*}
\begin{cases}
-M\biggl(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\biggr)\operatorname{div}
\biggl(|\nabla u|^{p(x)-2}\nabla u\biggr)=\lambda |u|^{r(x)-2}u +f(x,u)
&\text{in }\Omega,
\\
u=0 &\text{on }\partial\Omega,
\end{cases}
\end{align*}
where
$\Omega$
is a bounded domain of
$\mathbb{R}^N (N\geq 2)$
with smooth boundary
$\partial\Omega$, $\lambda>0$,
and $p$ and $r$,
are two continuous functions in
$\overline{\Omega}$.
Using variational methods combined with some properties of the
generalized Sobolev spaces, under appropriate assumptions on
$f$
and
$M$,
we obtain a
number of results on the existence of solutions.
In addition, we show the existence of
infinitely many solutions in the case when
$f$
satisfies the evenness condition.
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